Assume \(W=0\) . Then the semi-orthogonal decomposition follows from Propositions REF  and REF .
Assume \(W\)  is arbitrary.
Let \((d_i,w_i)_{i=1}^k\)  be a partition of \((d,w)\) .
Let \(\mathbb {D}\)  be a subcategory of \(\otimes _{i=1}^k D^b\left(\mathcal {X}(d_i)\right)_{w_i}\)  on which \(p_{\underline{d}*}q_{\underline{d}}^*\)  is fully faithful. Then
\(\text{MF}\left(p_{\underline{d}*}q_{\underline{d}}^*\,\mathbb {D}, W_d\right)\cong p_{\underline{d}*}\text{MF}\left(q^*_{\underline{d}}\,\mathbb {D}, p_{\underline{d}}^*W_d\right)\cong p_{\underline{d}*}q_{\underline{d}}^*\,\text{MF}\left(\mathbb {D}, \oplus _{i=1}^kW_{d_i}\right)\\\cong \text{MF}\left(\mathbb {D}, \oplus _{i=1}^kW_{d_i}\right).\) 
The semi-orthogonal decomposition for \(\text{MF}(\mathcal {X}(d), W)\)  follows from the semi-orthogonal decomposition for \(W=0\)  and [1]}. The argument for \(\text{MF}^{\text{gr}}(\mathcal {X}(d), W)\)  is similar and follows from [1]}.
